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Multivariable-Calculus · Partial-Derivatives

Gradient of y - 1/(x^2y)

y - \frac{1}{x^{2} y}

Step by step

  1. f(x, y) = y - \frac{1}{x^{2} y}

    The gradient is the vector of partial derivatives. Differentiate with respect to each variable, holding the others constant.

  2. \frac{\partial f}{\partial x} = \frac{2}{x^{3} y}

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = 1 + \frac{1}{x^{2} y^{2}}

    Treat every variable except y as a constant.

  4. \nabla f = \left[\begin{matrix}\frac{2}{x^{3} y}\\1 + \frac{1}{x^{2} y^{2}}\end{matrix}\right]

    Assemble the gradient vector. It points in the direction of steepest ascent.

Reveal the answer
\nabla f = \left[\begin{matrix}\frac{2}{x^{3} y}\\1 + \frac{1}{x^{2} y^{2}}\end{matrix}\right]